Reduced Basis Method for the Stokes Equations in Decomposable Parametrized Domains Using Greedy Optimization

Flow simulations in pipelined channels and several kinds of parametrized configurations have a growing interest in many life sciences and industrial applications. Applications may be found in the analysis of the blood flow in specific compartments of the circulatory system that can be represented as a combination of few deformed vessels from reference ones, e.g. pipes. We propose a solution approach that is particularly suitable for the study of internal flows in hierarchical parametrized geometries. The main motivation is for applications requiring rapid and reliable numerical simulations of problems in domains involving parametrized complex geometries. The classical reduced basis (RB) method is very effective to address viscous flows equations in parametrized geometries (see, e.g., [10]). An interesting alternative foresees a combination of RB with a domain decomposition approach. In this respect, preliminary efforts to reduce the global parametrized problem to local ones have led to the introduction of the so-called reduced basis element method to solve the Stokes problem [6], and more recently to the reduced basis hybrid method [3] and to the static condensation method [7]. In general, we are interested in defining a method able to maintain the flexibility of dealing with arbitrary combinations of subdomains and several geometrical deformations of the latter. A further new contribution to this field is the computation of the reduced basis functions through an optimization greedy algorithm

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references

A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems

The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations (Fink & Rheinboldt (1983), Noor & Peters (1980), Prud’homme et al. (2002)), and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of subdomains that are similar to a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation, but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual domain. We extend earlier work (Maday & Rønquist (2002), Maday & Rønquist (2004)) in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to constructing the basis functions, to the mapping of the velocity fields, to satisfying the inf-sup condition, and to 'gluing' the local solutions together in the multidomain case (Belgacem et al. (2000)). We also demonstrate an algorithm for choosing the most efficient precomputed solutions. Two-dimensional examples are presented for pipes, bifurcations, and couplings of pipes and bifurcations in order to simulate hierarchical flow systems

A REDUCED BASIS ELEMENT METHOD FOR COMPLEX FLOW SYSTEMS

Key words: reduced basis, domain decomposition, a posteriori error estimators, transfinite interpolation, empirical interpolation Abstract. The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations within domains belonging to a certain class. The method takes its roots in domain decomposition methods and reduced basis discretizations. 1–3 The basic idea is to first decompose the computational domain into smaller blocks that are topologically similar to a few reference shapes (or generic computational parts). Associated with each reference shape are precomputed solutions corresponding to the same governing partial differential equation, and similar boundary conditions, but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. 4,5 The approximation corresponding to the computational domain is then taken to be a linear combination of the precomputed solutions, mapped from the reference shapes for the different blocks to the actual domain. The variation of the geometry induces non-affine parameter dependence, and we apply the empirical interpolation technique to achieve a